Euclid may have been the first to give
a proof that there are infintely many primes. Below we give another proof by Filip Saidak [Saidak2005], similar to Goldbach's argument, but in a way even simpler.
There are infinitely many primes.
Let n > 1 be a positive integer. Since n and n+1 are consecutive
integers, they must be coprime, and hence the number N2 = n(n + 1) must have at
least two different prime factors. Similarly, since the integers n(n+1) and n(n+1)+1
are consecutive, and therefore coprime, the number N3 = n(n + 1)[n(n + 1) + 1]
must have at least 3 different prime factors. This can be continued indefinitely.